// xcas version=0.6.3 fontsize=20 font=0 // fltk 7Fl_Tile 13 -344 909 520 20 0 [ // fltk N4xcas6FigureE 13 -344 909 519 20 0 landscape=0 history=0.34103 geo=0.56876 mouse_param=0.090209 // fltk N4xcas12History_PackE 15 -1154 290 1329 20 0 [ // fltk 7Fl_Tile 37 -1154 268 101 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -1154 268 51 20 0 O:=point(0,0,0,£display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -1103 268 1 20 0 , // fltk 9Fl_Scroll 37 -1102 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -1102 277 27 20 0 pnt(pnt[point[0,0,0],2097152,"O"]) , // fltk 12Fl_Scrollbar 37 135 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 106 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -1051 268 98 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -1051 268 48 20 0 A:=point(1,0,0,£display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -1003 268 1 20 0 , // fltk 9Fl_Scroll 37 -1002 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -1002 275 27 20 0 pnt(pnt[point[1,0,0],2097152,"A"]) , // fltk 12Fl_Scrollbar 37 167 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 138 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -951 268 98 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -951 268 48 20 0 B:=point(0,1,0,£display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -903 268 1 20 0 , // fltk 9Fl_Scroll 37 -902 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -902 275 27 20 0 pnt(pnt[point[0,1,0],2097152,"B"]) , // fltk 12Fl_Scrollbar 37 164 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 135 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -851 268 98 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -851 268 48 20 0 C:=point(0,0,1,£display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -803 268 1 20 0 , // fltk 9Fl_Scroll 37 -802 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -802 276 27 20 0 pnt(pnt[point[0,0,1],2097152,"C"]) , // fltk 12Fl_Scrollbar 37 156 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 127 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -751 268 101 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -751 268 51 20 0 T:=tetrahedron(O,A,B,C,£display=cyan) , // fltk N4xcas10Log_OutputE 37 -700 268 1 20 0 , // fltk 9Fl_Scroll 37 -699 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -699 1600 27 20 0 pnt(pnt[polyhedron[group[point[0,0,0],point[1,0,0],point[0,1,0]],group[point[0,0,0],point[0,1,0],point[0,0,1]],group[point[0,0,0],point[1,0,0],point[0,0,1]],group[point[1,0,0],point[0,1,0],point[0,0,1]]],6,"T"]) , // fltk 12Fl_Scrollbar 37 573 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 544 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -648 268 80 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -648 268 30 20 0 I:=midpoint(A,B,display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -618 268 1 20 0 , // fltk 9Fl_Scroll 37 -617 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -617 298 27 20 0 pnt(pnt[point[1/2,1/2,0],2097152,"I"]) , // fltk 12Fl_Scrollbar 37 566 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 537 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -566 268 80 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -566 268 30 20 0 AC:=segment(A,C) , // fltk N4xcas10Log_OutputE 37 -536 268 1 20 0 , // fltk 9Fl_Scroll 37 -535 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -535 385 27 20 0 pnt(pnt[group[point[1,0,0],point[0,0,1]],0,"AC"]) , // fltk 12Fl_Scrollbar 37 563 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 534 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -484 268 58 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -484 268 30 20 0 assume(t=[0.32,0,1,0.01]) , // fltk N4xcas10Log_OutputE 37 -454 268 1 20 0 , // fltk N4xcas10Gen_OutputE 37 -453 268 27 20 0 parameter(t,0.0,1.0,0.32,0.01) ] , // fltk 7Fl_Tile 37 -424 268 98 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -424 268 48 20 0 M:=element(AC,t,£display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -376 268 1 20 0 , // fltk 9Fl_Scroll 37 -375 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -375 661 27 20 0 pnt(pnt[point[1-t,0,t],[2097152,[pnt(pnt[group[point[1,0,0],point[0,0,1]],0]),t]],"M"]) , // fltk 12Fl_Scrollbar 37 566 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 537 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -324 268 101 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -324 268 51 20 0 P:=orthogonal(I,line(I,M),£display=vert) , // fltk N4xcas10Log_OutputE 37 -273 268 1 20 0 , // fltk 9Fl_Scroll 37 -272 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -272 507 27 20 0 pnt(pnt[hyperplan([point[1-t-1/2,-1/2,t],point[1/2,1/2,0]]),2,"P"]) , // fltk 12Fl_Scrollbar 37 175 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 146 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -221 268 98 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -221 268 48 20 0 inter(T,evalf(P),£display=rouge+epaisseur_ligne_5) , // fltk N4xcas10Log_OutputE 37 -173 268 1 20 0 , // fltk 9Fl_Scroll 37 -172 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -172 1921 27 20 0 [pnt(pnt[group[point[0.5,0.5,0.0],point[0.0,0.32,0.0]],262145]),pnt(pnt[group[point[0.0,0.32,0.0],point[0.0,0.585365853659,0.414634146341]],262145]),pnt(pnt[group[point[0.5,0.5,0.0],point[0.0,0.585365853659,0.414634146341]],262145])] , // fltk 12Fl_Scrollbar 37 275 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 246 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -121 268 98 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -121 268 48 20 0 N:=single_inter(P,line(O,B),£display=epaisseur_point_5) , // fltk N4xcas10Log_OutputE 37 -73 268 1 20 0 , // fltk 9Fl_Scroll 37 -72 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 -72 413 27 20 0 pnt(pnt[point[0,1-t-1/2/2-1/4/(-1/2),0],2097152,"N"]) , // fltk 12Fl_Scrollbar 37 375 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 346 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 -21 268 101 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 -21 268 51 20 0 MN:=segment(M,N,£display=epaisseur_ligne_5) , // fltk N4xcas10Log_OutputE 37 30 268 1 20 0 , // fltk 9Fl_Scroll 37 31 268 49 20 0 [ // fltk N4xcas10Gen_OutputE 37 31 570 27 20 0 pnt(pnt[group[point[1-t,0,t],point[0,-2*(1-t-1/2/2-1/4),0]],262144,"MN"]) , // fltk 12Fl_Scrollbar 37 60 268 20 20 0 [] , // fltk 12Fl_Scrollbar 305 31 16 29 20 0 [] ] ] , // fltk 7Fl_Tile 37 82 268 58 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 82 268 30 20 0 evalf(distance(M,N)) , // fltk N4xcas10Log_OutputE 37 112 268 1 20 0 , // fltk N4xcas10Gen_OutputE 37 113 268 27 20 0 0.816823114266 ] , // fltk 7Fl_Tile 37 142 268 31 20 0 [ // fltk N4xcas19Multiline_Input_tabE 37 142 268 30 20 0 , // fltk N4xcas10Log_OutputE 37 172 268 1 20 0 ] ] // fltk N4xcas5Geo3dE 323 -318 517 493 20 0 -0.2777,1.2777,-0.2777,1.2777,[pnt(pnt[point[0,0,0],2097152,"O"]),pnt(pnt[point[1,0,0],2097152,"A"]),pnt(pnt[point[0,1,0],2097152,"B"]),pnt(pnt[point[0,0,1],2097152,"C"]),pnt(pnt[polyedre[group[point[0,0,0],point[1,0,0],point[0,1,0]],group[point[0,0,0],point[0,1,0],point[0,0,1]],group[point[0,0,0],point[1,0,0],point[0,0,1]],group[point[1,0,0],point[0,1,0],point[0,0,1]]],6,"T"]),pnt(pnt[point[1/2,1/2,0],2097152,"I"]),pnt(pnt[group[point[1,0,0],point[0,0,1]],0,"AC"]),parameter(t,0.0,1.0,0.32,0.01),pnt(pnt[point[1-t,0,t],[2097152,[pnt(pnt[group[point[1,0,0],point[0,0,1]],0]),t]],"M"]),pnt(pnt[hyperplan([point[1-t-1/2,-1/2,t],point[1/2,1/2,0]]),2,"P"]),group[pnt(pnt[group[point[0.5,0.5,0.0],point[0.0,0.32,0.0]],262145]),pnt(pnt[group[point[0.0,0.32,0.0],point[0.0,0.585365853659,0.414634146341]],262145]),pnt(pnt[group[point[0.5,0.5,0.0],point[0.0,0.585365853659,0.414634146341]],262145])],pnt(pnt[point[0,1-t-1/2/2-1/4/(-1/2),0],2097152,"N"]),pnt(pnt[group[point[1-t,0,t],point[0,-2*(1-t-1/2/2-1/4),0]],262144,"MN"]),0.816823114266],-0.2777,1.2777,0.58134,-0.68925,0.27849,0.33078,0.05,0.05,1,2097152,1,1.8,0,1,65,[[0,0,1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,-1,0,0,180,1,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,180,1,0,0,0]],24,18,256,0,100,0,0,1,0.1 , // fltk N4xcas10Log_OutputE 13 175 909 1 20 0 ] , // fltk 7Fl_Tile 13 178 909 31 20 0 [ // fltk N4xcas23Comment_Multiline_InputE 13 178 909 30 20 0 Ici on peut faire calculer par xcas les coordonnees de M et N, admises dans l'enonce , // fltk N4xcas10Log_OutputE 13 208 909 1 20 0 ] , // fltk 7Fl_Tile 13 211 909 54 20 0 [ // fltk N4xcas19Multiline_Input_tabE 13 211 909 30 20 0 simplifier(coordinates(M),coordinates(N)) , // fltk N4xcas10Log_OutputE 13 241 909 1 20 0 , // fltk N4xcas8EquationE 13 242 909 23 20 0 [-t+1,0,t],[0,t,0] ] , // fltk 7Fl_Tile 13 267 909 68 20 0 [ // fltk N4xcas19Multiline_Input_tabE 13 267 909 30 20 0 eq:=simplifier(distance(M,N)) , // fltk N4xcas10Log_OutputE 13 297 909 1 20 0 , // fltk N4xcas8EquationE 13 298 909 37 20 0 sqrt(3*t^2-2*t+1) ] , // fltk 7Fl_Tile 13 337 909 102 20 0 [ // fltk N4xcas19Multiline_Input_tabE 13 337 909 30 20 0 factoriser(deriver(eq,t)) , // fltk N4xcas10Log_OutputE 13 367 909 1 20 0 , // fltk N4xcas8EquationE 13 368 909 71 20 0 ((3*t-1)*sqrt(3*t^2-2*t+1))/(3*t^2-2*t+1) ] , // fltk 7Fl_Tile 13 441 909 31 20 0 [ // fltk N4xcas23Comment_Multiline_InputE 13 441 909 30 20 0 donc le minimum est atteint en t=1/3 et vaut , // fltk N4xcas10Log_OutputE 13 471 909 1 20 0 ] , // fltk 7Fl_Tile 13 474 909 82 20 0 [ // fltk N4xcas19Multiline_Input_tabE 13 474 909 30 20 0 substituer(eq,t=1/3) , // fltk N4xcas10Log_OutputE 13 504 909 1 20 0 , // fltk N4xcas8EquationE 13 505 909 51 20 0 sqrt(2/3) ] , // fltk 7Fl_Tile 13 558 909 54 20 0 [ // fltk N4xcas19Multiline_Input_tabE 13 558 909 30 20 0 evalf(sqrt(2/3)) , // fltk N4xcas10Log_OutputE 13 588 909 1 20 0 , // fltk N4xcas8EquationE 13 589 909 23 20 0 0.816496580928 ] , // fltk 7Fl_Tile 13 614 909 31 20 0 [ // fltk N4xcas19Multiline_Input_tabE 13 614 909 30 20 0 , // fltk N4xcas10Log_OutputE 13 644 909 1 20 0 ]